It's similarly impossible to build roads with enough capacity for everyone to drive. Cars work fine in small towns and low-density areas, but once the population and/or density grows too high, traffic becomes terrible. If you fix the population density and let the city grow, the traffic that needs to cross any particular cross section of the city grows asymptotically faster than the length of the cross section (and therefore road capacity). The traditional solution was lowering the density by replacing houses with roads, but that's no longer popular.
Traffic is a hard problem. There is no single solution that fits everywhere, and you will never be able to go everywhere with your preferred means of transit. At least if you are not a major political leader such as the US president. If you are, you can just stop the normal traffic and let the peasants wait.
Sure. Trains in very dense areas with homogenous travel patterns (e.g, commuting to work) and roads everywhere else. Essentially what we have today in many places.
Of course, commuting to sit in front of a computer all day is pretty absurd. Maybe unnecessary commuting is the real problem.
Traffic is a hard problem. There is no single solution that fits everywhere, and you will never be able to go everywhere with your preferred means of transit. At least if you are not a major political leader such as the US president. If you are, you can just stop the normal traffic and let the peasants wait.